Ordered Optimization Professional Playbook

Ordered optimization should feel like shrinking or finalizing candidates because the order gives a proof.

binary search boundary:

index:      0   1   2   3   4   5   6
predicate:  F   F   F   T   T   T   T
                         ^
                         first true answer

The search is for the boundary, not for a lucky working sample.

Dijkstra heap finality:

heap before pop:

cost 2: A   <-- cheapest unsettled state
cost 5: B
cost 8: C

pop A:
    A is final only because all edge weights are nonnegative.

greedy interval exchange:

candidate intervals sorted by finish:

    [----A----]
       [--B--]
             [---C---]

Pick the compatible interval that finishes earliest.
It leaves the widest future suffix for the rest of the solution.

sweep line active set:

time:     1      3      5      7
events:  +A     +B     -A     -B

active after event:
         {A}   {A,B}   {B}    {}

Queries at time t read exactly the active set after all events <= t.

top-k heap:

stream:  9  1  7  3  8     k = 3

kept heap after scan:
        7
       / \
      9   8

The root is the weakest kept candidate; anything smaller cannot enter top k.

binary search:

left                 mid                 right
 v                   v                    v
[ ? ][ ? ][ ? ][ x ][ ? ][ ? ][ ? ][ ? ]

compare at mid, then discard one side by proof.
invariant: the answer remains inside the undiscarded side

binary search on answer:

candidate answer:
1   2   3   4   5   6   7   8
F   F   F   F   T   T   T   T
                ^
              boundary

The ordered thing is the answer value itself.
invariant: feasibility is monotonic across candidate values

Dijkstra:

start --2--> A --4--> target
  |                         ^
  +----7----> B -----1------+

heap order:
cost 2: A
cost 7: B

The cheapest unsettled cost is processed first.
invariant: popped cost is final when all edges are nonnegative

interval greedy:

time:  0    2    4    6    8
       [----A----]
          [--B--]
                [--C--]

Sort by the endpoint that makes the safe local choice provable.
invariant: the chosen interval leaves an optimal-compatible future

heap scheduling:

tasks arrive:
T1 duration 4
T2 duration 2
T3 duration 3

machine heap by next free time:
time 0: M1, M2
time 2: M2
time 4: M1

The top resource is the next available one.
invariant: heap top is the resource that can be reused earliest

sweep line:

coordinate:  1      3      5      8
event:      +A     +B     -A     -B

active:
after 1: {A}
after 3: {A, B}
after 5: {B}
after 8: {}

invariant: active set contains exactly events covering this coordinate

order statistics:

values:       40   10   30   10
sorted uniq:  10   30   40
rank:          1    2    3

rank keeps order while making indexes dense.
invariant: rank comparisons match original value comparisons

The order must connect directly to the proof. Sorting intervals by start time helps merging; sorting by finish time helps maximum compatible selection.

Binary search on answer requires a monotonic predicate:

false false false true true true

or the reverse. The predicate must answer a yes/no question for one candidate value.

Examples:

The returned answer is the boundary, not the first value that happened to work in a trial loop.

Greedy sorting needs an exchange idea:

If an optimal solution does not use my local choice, I can swap my local choice
into it without making the solution worse.

This is why earliest-finish interval selection works and arbitrary “take the largest-looking item” often fails. The local choice must preserve the future search space or objective value.

identify ordered domain
define the safe decision invariant
choose bounds, heap, sorted events, or ordered structure

while undecided candidates remain:
    inspect the next ordered candidate
    prove which candidates are final, discarded, or still active
    update the maintained structure
    update answer boundary or best known result

return the boundary, optimum, order, or active-query result

Trace 1: binary search on answer.

capacity candidates:
10  11  12  13  14  15
F   F   F   T   T   T

step 1:
    mid = 12, feasible = false
    discard [10, 11, 12]
    invariant: no smaller capacity can work if 12 fails

step 2:
    mid = 14, feasible = true
    keep [13, 14], discard bigger only after narrowing
    invariant: answer is still the first feasible boundary

Trace 2: Dijkstra heap.

heap:
    [2,A] [5,B] [9,C]

step 1:
    pop [2,A]
    commit dist[A] = 2
    invariant: A is final because no unsettled path can be cheaper

step 2:
    relax A -> D with +3
    push [5,D]
    invariant: heap contains known unsettled candidates ordered by distance

Use this family when order proves work can be skipped. Be careful when:

The clean test is: can you point to the exact ordering proof that makes ignored candidates irrelevant? If not, the pattern choice is not ready.

Invariant:

The answer is always inside [left, right].

def search_insert(nums, target):
    left = 0
    right = len(nums)

    while left < right:
        mid = (left + right) // 2
        if nums[mid] < target:
            left = mid + 1
        else:
            right = mid

    return left

Invariant:

Shooting at the earliest ending balloon preserves the most room for future
balloons.

def find_min_arrow_shots(points):
    points.sort(key=lambda x: x[1])
    arrows = 0
    arrow_end = None

    for start, end in points:
        if arrow_end is None or start > arrow_end:
            arrows += 1
            arrow_end = end

    return arrows

Binary search maintains a boundary.

Lower-bound form:

left is the first unknown candidate
right is one past the last unknown candidate

At the end, left == right is the insertion point or first satisfying index.

Greedy is not “take what looks good.” It needs an exchange argument.

Common safe orders:

Input:

nums = [7, 2, 5, 10, 8]
k = 2

Bounds:

left = 10
right = 32

Feasibility examples:

limit 21 works: [7,2,5] [10,8]
limit 15 fails: [7,2,5] [10] [8]

The search converges to 18.

Intervals:

[1, 4], [2, 5], [7, 9]

Events:

1 start -> active 1
2 start -> active 2
4 end   -> active 1
5 end   -> active 0
7 start -> active 1
9 end   -> active 0

Maximum overlap:

2